Accurate Numerical Simulations Research Articles

A data-free Kolmogorov–Arnold Network-based method for structured mesh generation

Mesh generation is a critical but time-consuming process for stable and accurate numerical simulations. Although multi-layer perceptron-based meshing methods can be effective, they suffer from slow training convergence and heavy reliance on prior datasets. To overcome these problems, we propose the Kolmogorov–Arnold Network-based meshing network, an efficient data-free method for structured mesh generation. The proposed method takes the meshing task as an optimization problem and embeds meshing-related differential equations into the loss function of Kolmogorov–Arnold Networks. It employs two parts to generate meshes efficiently. The Kolmogorov–Arnold Network part introduces learnable activation functions on the edges of the network, which enables the network to learn meshing rules between parametric and computational domains. The physics-informed learning part provides meshing-related information to guide the network training. Finally, the proposed method can produce high-quality structured meshes with a user-defined number of quadrilateral or hexahedral cells through feed-forward prediction. Experiments on different geometries show that the proposed method achieves up to three orders of magnitude improvement in meshing efficiency compared to traditional methods. It also outperforms state-of-the-art multi-layer perceptron-based methods, yielding high-quality meshes in both two-dimensional and three-dimensional cases without prepared data.

Mechanical performance analysis method for ribbed H-section aluminum alloy members with initial curvature and torsion angle

In this study, an experimental investigation was conducted on the axial compression performance of ribbed H-section aluminum alloy members with initial curvature and torsion angle under varying boundary conditions, including one end hinged with the other rigidly connected, and both ends rigidly connected. Ultimate bearing capacity and failure modes were identified under real loads and subsequently compared with previous findings from our research group on members with hinged ends. To account for initial imperfections introduced during processing and transportation, 3D scanning technology was utilized to capture the precise geometrical dimensions, constructing an accurate numerical simulation model. The experimental results were corroborated with numerical simulations, leading to the proposal of an analytical method for members with initial curvature and torsion angle. Furthermore, extensive parametric analysis elucidated the impact of initial curvature, torsion angle, and slenderness ratio on the ultimate bearing capacity, culminating in the formulation of the stability factor and calculated length factor based on numerical outcomes. The study discovered significant variances in bearing capacity under different boundary conditions, with one-end hinged and one-section rigidly connected, and two-end rigidly connected conditions exhibiting 1.4 and 2.1 times the capacity of the hinged-at-both-ends scenario. Under different boundary conditions, the axial compression members were subjected to flexural-torsional buckling failure. Moreover, when the ultimate bearing capacity was reached, the lower flange of the member and the web near the lower flange appeared obvious buckling phenomenon. The numerical analysis aligned well with experimental data, validating the simulation method's reliability and revealing the stress distribution and evolution during member failure. These findings offer vital theoretical insights and technical support for engineering design and practical applications.

Advancing superconductivity research: Insights from numerical simulations of potassium fullerenide-60 and gold and with Ginzburg-Landau theory

There are many types of superconductors, including gold ormus and some fullerene derivatives. Gold can become a superconductor at extremely low temperatures (<1 K), allowing it to conduct electricity without resistance. While not as commonly used as materials like niobium or lead, gold superconductors are valuable for research and development in superconductivity. Fullerene derivatives like potassium fullerenide-60 also exhibit high superconductivity. Limited studies have been conducted on both gold ormus and superconducting fullerene derivatives. Our study of numerical simulations of the Ginzburg-Landau theory in superconductors for gold ormus and potassium fullerenide-60 has yielded important results. We have successfully simulated class-I and class-II superconducting gold ormus, as well as potassium fullerenide-60, using the Runge-Kutta fourth order method. Our analysis demonstrates the convergence of our simulation outcomes and highlights the importance of considering truncation error and selecting appropriate step sizes for accurate results. The periodic factor of penetration (PFP) for each superconductor has been determined, with class-I superconducting gold having a PFP of 250 nm, class-II superconducting gold having a PFP of 566.2 nm, and potassium fullerenide-60 having a PFP of 1.374 nm. Additionally, our study reveals the relationship between the periodic penetration factor and the length of the penetration depth, showing that the PFP reaches a minimum value at a penetration depth length of 130 nm. Overall, our findings contribute to a better understanding of superconductivity in gold ormus and potassium fullerenide-60, emphasizing the importance of accurate numerical simulations for studying complex physical phenomena. Our study confirmed the accuracy of the Runge-Kutta fourth-order method in simulating superconductors. By examining the PFP for various superconducting materials, we identified trends in penetration depth, shedding light on superconductivity. Our simulations give valuable insights for advancing research in this field, with the Runge-Kutta fourth-order method striking a balance between accuracy and efficiency. Careful parameter adjustment ensures reliable simulations and contributes to progress in superconductivity research.

Research on the Hydrodynamic Characteristics of a Rectangular Otter Board in Different Work Postures Based on a Dynamic Model

This paper investigates the hydrodynamic characteristics of a rectangular otter board in different working postures by using a dynamic model. Dynamic models are mainly based on dynamic mesh techniques. The results of the dynamic model are, compared to the model test, carried out in a flume tank. Furthermore, different rotation speeds of dynamic model were analyzed. The research results are as follows: compared to flume tank results, the maximum error of the dynamic model is 23.77%. Moreover, the influence of rotation speed on the hydrodynamic board is not obvious, and 2 deg./s was chosen as the rotation speed. When the board is tilted slightly (including four working postures), its lift-to-drag ratio first increases slightly and then gradually decreases. Compared with the other three working postures, the pressure center coefficient of the board does not change significantly when it is tilted inward. When studying different working angles (including AOA and tilt angle) of the otter board, the numerical dynamic model significantly reduces repetitive setup work, making simulations more efficient. Its ability to provide continuous curves and a large volume of results offers researchers a more detailed and comprehensive understanding of the board’s hydrodynamics. Additionally, the dynamic model supports innovative fishery equipment development by allowing more accurate and continuous numerical simulations.

Integration of Geological Model and Numerical Simulation Technique to Characterize the Remaining Oil of Fractured Biogenic Limestone Reservoirs

AbstractFine geological modeling leads to accurate reservoirs numerical simulations. Fractured biogenic limestone has abundant storage spaces and flow paths to accumulate oil and gas. The complexity and diversity of fractured biogenic limestone also lead to challenges in accurately characterizing its pore volume and remaining oil. This investigation aimed to enhance the understanding of fractured biotite reservoir properties via geological modeling. Numerical simulations were used to characterize the remaining oil during the late stage of field development. Considering the differences in porosity and permeability between fractures and matrix, a facies-controlled stochastic modeling technique was used to establish a dual-porosity and dual-permeability (DPDP) model for numerical simulation. Core information, logging data, and multiple seismic attributes were combined to guide low-level sequence fault interpretation for tectonic refinement. Based on classified seismic inversion, sedimentary phases were reconstructed. A discrete fracture network (DFN) model was obtained based on fracture occurrences and density models. The optimized discrete adjoint (ODA) algorithm was utilized to calibrate model parameters. The findings revealed that dense tectonic fractures develop in thick biogenic limestone areas. Combined with advanced reservoir simulation technology, these findings suggest that areas of thicker biogenic limestone were consistent with areas of higher fracture matrix conductivity multipliers. The remaining oil distribution patterns were investigated, and to deploy new wells was guided. Therefore, it is essential to better understand the tectonic characteristics of fractured biogenic limestone reservoirs and their remaining oil distribution patterns by integrating multiple sources of information and mastering advanced reservoir simulation technology for oilfield development.

A Composite Approach for Evaluating Operational Cloud Seeding Effect in Stratus Clouds

Robust water management is in intense demand in many water scarcity areas, such as arid and semi-arid regions in the world. As part of the regional water management strategy, rain enhancement is vital to replenish groundwater reservoirs, and the key challenge is how to assess its effectiveness. Some recent weather modification experiments attained cloud seeding effect through advanced in situ measurement coupled with accurate numerical simulation. However, there is still a lack of an objective and scientific approach to quantitatively evaluate the rain enhancement effect, especially for many non-randomized operational cloud seeding activities in China. In this study, we proposed a composite evaluation approach by analyzing two operational aircraft cloud seeding cases in stratus clouds in Shaanxi, China. By calculating the aircraft cloud seeding agent plumes, the target areas (as well as the control areas) of cloud seeding were dynamically and roughly determined. Physical properties, such as radar reflectivity and precipitation, were individually quantified in these areas. The cloud seeding effect was then evaluated by calculating the difference in parameter variation between target and control areas. This approach can be applied to qualitative analysis in a single aircraft cloud seeding operation and can also provide quantitative statistical results from multiple cloud seeding cases. We found that the average precipitation enhancement percentage of 18 operational aircraft cloud seeding cases is ~4.84%. Note that the homogeneity hypothesis of the seeding cloud, the error in the calculation of the target area, and the selection of control areas are the major uncertainties likely in the evaluation of the cloud seeding effect by this approach.

Development of an improved mass transfer model for condensation with dynamic feedback regulation

In the field of condensation research, accurate and efficient numerical simulation models are highly desired. One of the widely used phase change mass transfer models is known as the Lee model. However, this model has limitations due to uncertainties in the mass transfer intensity factor, which can lead to simulation failures and impede large-scale engineering applications. This study presents an improved mass transfer model which employs a tuning function to dynamically and accurately modify the mass transfer intensity factor in each two-phase cell via feedback adjustment. This approach enables immediate and customized adjustment of the factor for each grid within the limit of the tuning function, significantly reducing the errors caused by repeated artificial adjustments and related uncertainties. The adjustment capabilities of the hyperbolic tangent and softsign functions are evaluated. The softsign function is superior in both regulation efficiency and accuracy. The model's accuracy is verified by comparing with previous experiments and simulations, as well as by the one-dimensional Stefan problem. The proposed model achieves high prediction accuracy, reduces the risk of simulation dispersion, and significantly enhances computational efficiency with the iteration number reduced by nearly half under certain conditions compared to the Lee model. Furthermore, the new model shows robustness as the prediction accuracy is insensitive to variations in critical parameters. The improved model is expected to provide valuable assistance for future research and applications in flow condensation with outstanding computational accuracy, efficiency, and stability.

Effect of infusion direction on convection-enhanced drug delivery to anisotropic tissue.

Convection-enhanced delivery (CED) can effectively overcome the blood-brain barrier by infusing drugs directly into diseased sites in the brain using a catheter, but its clinical performance still needs to be improved. This is strongly related to the highly anisotropic characteristics of brain white matter, which results in difficulties in controlling drug transport and distribution in space. In this study, the potential to improve the delivery of six drugs by adjusting the placement of the infusion catheter is examined using a mathematical model and accurate numerical simulations that account simultaneously for the interstitial fluid (ISF) flow and drug transport processes in CED. The results demonstrate the ability of this direct infusion to enhance ISF flow and therefore facilitate drug transport. However, this enhancement is highly anisotropic, subject to the orientation of local axon bundles and is limited within a small region close to the infusion site. Drugs respond in different ways to infusion direction: the results of our simulations show that while some drugs are almost insensitive to infusion direction, this strongly affects other compounds in terms of isotropy of drug distribution from the catheter. These findings can serve as a reference for planning treatments using CED.

Numerical simulation of two-phase oil–water flow in fractured-vuggy reservoirs based on the coefficient of porous medium proportion and coupled regions

Based on the flow characteristics of fluids in various reservoir media, fractured-vuggy oil reservoirs can be classified into seepage zones and conduit flow zones. An interface exists between these two regions, where the movement of formation fluid near this interface is characterized by a coupling or transitional phenomenon between seepage and conduit flow. However, the complexity of this coupling interface poses challenges for traditional numerical simulations in accurately representing the intricate fluid dynamics within fractured-vuggy oil reservoirs. This limitation impacts the development planning and production adjustment strategies for fractured-vuggy oil reservoirs. Consequently, achieving accurate characterization and numerical simulation of these systems remains a critical challenge that requires urgent attention. A new mathematical model for oil-water two-phase flow in fractured-vuggy oil reservoirs is presented, which developed based on a novel coupling method. The model introduces the concept of the proportion coefficient of porous media within unit grids and defines a coupling region. It employs an enhanced Stokes–Brinkman equation to address the coupling issue by incorporating the proportion coefficient of porous media, thereby facilitating a more accurate description of the coupling interface through the use of the coupling region. Additionally, this proportion coefficient characterizes the unfilled cave boundary, simplifying the representation of model boundary conditions. The secondary development on the open-source fluid dynamics software is conducted by using matrix &amp; laboratory (MATLAB). The governing equations of the mathematical model are discretized utilizing finite volume methods and applying staggered grid techniques along with a semi-implicit calculation format for pressure coupling—the Semi-Implicit Method for Pressure Linked Equations algorithm—to solve for both pressure and velocity fields. Under identical mechanism models, comparisons between simulation results from this two-phase flow program and those obtained from Eclipse reveal that our program demonstrates superior performance in accurately depicting flow states within unfilled caves, thus validating its numerical simulation outcomes for two-phase flow in fractured cave reservoirs. Utilizing the S48 fault-dipole unit as a case study, this research conducted numerical simulations to investigate the water-in-place (WIP) behavior in fractured-vuggy oil reservoirs. The primary focus was on analyzing the upward trend of WIP and its influencing factors during production across various combinations of fractures and dipoles, thereby validating the feasibility of the numerical modeling approach in real-world reservoirs. The simulation results indicated that when multiple dissolution cavities at different locations communicated with the well bottom sequentially, the WIP in the production well exhibited a staircase-like increase. Furthermore, as the distance between bottom water and well bottom increased, its effect on water intrusion into the well diminished, leading to a slower variation in the WIP curve. These characteristics manifested as sudden influxes of water flooding, rapid increases in water levels, and gradual rises—all consistent with actual field production observations. The newly established numerical simulation method for fractured-vuggy oil reservoirs quantitatively describes two-phase flow dynamics within these systems, thus effectively predicting their production behaviors and providing guidance aimed at enhancing recovery rates typically observed in fractured-vuggy oil reservoirs.

Identification of hardening and fracture behaviors of titanium alloy by a hybrid digital image correlation and finite element method

The reliable identification of deformation and fracture behaviors of titanium alloys is essential for accurate numerical simulation in sheet metal forming, cutting process and crash. In this study, an inverse method is proposed to identify both strain hardening and fracture behavior of titanium alloys by a hybrid experimental-finite element analysis. Uniaxial and notched tensile specimens are tested on the additively manufactured and cold rolled sheet Ti6Al4V alloys. The distributions of displacement and strain on the specimen surface during the loading are measured by digital image correlation method. An identification strategy, considering the stress–strain curve, surface strain evolution and variation of specimen width, is proposed to identify the strain hardening model parameters at large strain range. Finite element simulations are performed to update the model parameters by using the iterative inverse method. The parameters of fracture model are determined through experiments on tensile specimens in combination with numerical simulation. The experimental and simulation results show the feasibility and effectiveness of the proposed identification method, that it can be further used to characterize the plastic and fracture behaviors of other metallic materials in practice.

Regularity and uniqueness of solutions for compressible non-newtonian MHD equations with variable electrical conductivity and energy effects

This paper examines the existence, uniqueness, and regularity of solutions for a system of compressible non-Newtonian magnetohydrodynamics (MHD) equations with variable electrical conductivity and considering the energy equation. We establish the necessary conditions on fluid properties, external forces, and initial and boundary conditions to ensure the well-posedness of the system. Our results demonstrate the existence of regular solutions in appropriate functional spaces, the uniqueness of these solutions, and their higher regularity under smooth initial and boundary conditions. These findings provide a mathematical baseline for the study of complex interactions in compressible non-Newtonian MHD flows, and particularly the findings can be considered to support more accurate numerical simulations concerning a physical situation as well as deeper analytical solutions.

Gravitational-Wave Data Analysis with High-Precision Numerical Relativity Simulations of Boson Star Mergers.

Gravitational-wave signals detected to date are commonly interpreted under the paradigm that they originate from pairs of black holes or neutron stars. Here, we explore the alternative scenario of boson-star signals being present in the data stream. We perform accurate and long (∼20 orbits) numerical simulations of boson-star binaries and inject the resulting strain into LIGO noise. Our Bayesian inference reveals that some boson-star signals are degenerate with current approximants, albeit with biased parameters, while others exhibit smoking-gun signatures leaving behind conspicuous residuals.

Numerical analysis of fractional‐order Euler–Bernoulli beam model under composite model

The primary objective of this study is to develop a new constitutive model by combining a fractional‐order Kelvin–Voigt model with an Abel dashpot element in parallel. Subsequently, this new model will be incorporated into the Euler–Bernoulli beam's governing equation, utilizing shifted Legendre polynomials as basis functions, a classical orthogonal polynomial system, to solve the fractional‐order partial differential equations. By comparing the numerical solutions with the analytical solutions, we aim to evaluate the applicability of shifted Legendre polynomials in solving such problems and the accuracy of the obtained numerical solutions. Furthermore, we will investigate the performance of viscoelastic HDPE beams under different loading conditions and conduct a comparative analysis of the displacements of HDPE beams under the new constitutive model and the traditional fractional‐order Kelvin–Voigt model. Through this research, we hope to gain a deeper understanding of the characteristics of fractional‐order phenomena and provide more accurate and efficient numerical simulation and analysis methods for the field of structural mechanics, promoting the development of related engineering applications.

Failure analysis of Nehbandan granite under various stress states and strain rates using a calibrated Riedel–Hiermaier–Thoma constitutive model

This study presents a comprehensive procedure for determining, calibrating, and validating the Riedel–Hiermaier–Thoma (RHT) material model parameters for granite. The process involves collecting a comprehensive dataset of conventional mechanical tests conducted on various types of granite worldwide. Based on this dataset, one set of RHT material model parameters is determined. Additionally, a specific granite sample from Iran, known as Nehbandan granite, is characterized through physical and mechanical testing to obtain another set of parameters. The challenging task of determining the third set of parameters, which are difficult to obtain analytically and experimentally, is accomplished through a calibration process that iteratively adjusts the parameters based on comparisons between numerical simulation results and experimental data. To validate the determined parameters, a series of tests, including uniaxial compressive strength (UCS), triaxial compressive strength (TCS), Brazilian tensile strength, and dynamic Brazilian using split Hopkinson pressure bar (SHPB) tests, are conducted on the Nehbandan granite. These tests are also simulated using LS-Dyna software, and the numerical simulation results are compared with the corresponding experimental test results. The comparison between the numerical and experimental data serves as a means of validating and verifying the accuracy and reliability of the determined RHT material model parameters for granite. The results demonstrate the successful determination and calibration of the RHT material model parameters for granite. The model exhibits effectiveness in predicting the behavior of granite under various loading conditions. The validation process confirms the accuracy and reliability of the determined parameters through a close agreement between numerical simulations and experimental data. The findings contribute to a better understanding of granite's mechanical response and provide a reliable tool for simulating and predicting its behavior in engineering applications. The validated RHT material model parameters offer a robust framework for accurate numerical simulations, enabling engineers to make informed decisions in rock engineering projects.

A comprehensive mathematical model considering multiple mechanisms for CO2 displacement of methane in shale nanopores

With the rise in energy demand, shale gas exploitation has gained prominence, particularly through the use of carbon dioxide (CO2) injection for enhanced gas recovery. However, the intricate gas flow mechanisms within this process in shale nano-pores remain poorly understood, which greatly limited the accurate numerical simulation of shale gas production by CO2 injection. This study investigates the flow mechanisms of CH4 and CO2 gases in shale nano-pores, developing an improved apparent relative permeability model for the two-component CH4–CO2 gas, incorporating multiple transport mechanisms. Sensitivity analysis includes factors such as pressure, pore radius, viscosity correction effects, flow mechanisms, and composition fractions. Key findings from this study are as follows: (1) Increasing pore radius or decreasing pressure enhances the apparent permeability of CH4–CO2 gas, with pressure effects becoming negligible above 20 MPa. (2) The impact of viscosity correction on gas permeability diminishes with higher pressure or larger pore radius, becoming negligible for pore radii over 10 nm. (3) Larger pore radii increase slip flow and Knudsen diffusion, while molecular and surface diffusion proportions decrease; surface diffusion and slip flow dominate at high pressures. (4) A higher CH4–CO2 component ratio increases CH4 permeability and decreases CO2 permeability, with minimal influence from pressure and pore radius. These results can improve numerical simulations for shale gas production by using CO2 injection.

Physical modeling of conjugate heat transfer for multiregion and multiphase systems with the Volume-of-Fluid method

AbstractThe Volume-of-Fluid (VOF) method is commonly used for numerical simulations of phase change phenomena, such as nucleate boiling or droplet evaporation. A key issue with the standard VOF method is the averaging of the liquid and vapor properties in interface cells, which causes non-physical conjugate heat transfer with a solid wall. Therefore, we aim at a physical model for conjugate heat transfer between a solid and a multiphase fluid. The first measure for higher quality simulations is the splitting of the single temperature field in the fluid region into separate liquid and vapor temperature fields. The second measure is the development of a new, more physical temperature boundary condition for conjugate heat transfer between a solid region and a multiphase fluid, based on experimental results, theoretical models and theoretical considerations. In interface cells, the vapor phase is excluded from the conjugate heat transfer because only heat transfer to the liquid phase occurs resp. dominates. Additionally, the conjugate heat transfer between solid and liquid in the interface cells is performed with virtual subcells, depending on the respective volume fraction of the liquid phase. This new approach (we name it distinctive approach) is successfully validated for energy conservation, and stability issues are discussed for the first time. Significant differences to simulations with averaged properties are observed due to the (now) physically correct modeling of conjugate heat transfer. In our boiling cases, the more accurate numerical simulations lead to considerably larger bubble growth rates. Higher quality simulations are also expected for nearly all applications, where there is a three-phase contact line, be it vapor bubbles in nucleate boiling or droplets impacting on a heated surface.

Advanced cyclic constitutive model and parameter calibration method for austenitic stainless steel

Austenitic stainless steel is recognized for its potential in improving structural seismic resistance due to its exceptional ductility. This study seeks to analyse the material characteristics and cyclic constitutive model of austenitic stainless steel under cyclic loading using a combination of experiments and numerical studies. The primary goal is to enhance the precision of numerical simulations predicting the seismic performance of stainless steel and to minimize the costs associated with calibrating the parameters of the cyclic constitutive model. By conducting cyclic loading experiments on stainless steel with different plate thicknesses, the hysteresis curves under different loading protocols were obtained. The loading protocol significantly influences the cyclic hardening behaviour of stainless steel. To calibrate material parameters in the Chaboche model more effectively, a new parameter calibration method based on genetic algorithm optimization is proposed. Additionally, a method is proposed to calibrate the material parameters of the Hu model by utilizing the monotonic tensile stress-strain curve of austenitic stainless steel, which provides a new material model option for the numerical simulation of stainless steel seismic performance. The applicability of the Chaboche model and the Hu model in numerical simulations of stainless steel members is then verified and compared using existing seismic performance test results. The findings indicate that the Hu model offers more accurate numerical simulations of cyclic loading on austenitic stainless steel members compared to the Chaboche model.

Closed-loop designed open-loop control of quantum systems: An error analysis

Quantum Lyapunov control, an important class of quantum control methods, aims at generating converging dynamics guided by Lyapunov-based theoretical tools. However, unlike the case of classical systems, disturbance caused by quantum measurement hinders direct and exact realization of the theoretical feedback dynamics designed with Lyapunov theory. Regarding this issue, the idea of closed-loop designed open-loop control has been mentioned in literature, which means to design the closed-loop dynamics theoretically, simulate the closed-loop system, generate control pulses based on simulation and apply them to the real plant in an open-loop fashion. Based on bilinear quantum control model, we analyze in this article the error, i.e., difference between the theoretical and real systems’ time-evolved states, incurred by the procedures of closed-loop designed open-loop control. It is proved that the error at an arbitrary time converges to a unitary transformation of initialization error as the number of simulation grids between 0 and that time tends to infinity. Moreover, it is found that once the simulation accuracy reaches a certain level, adopting more accurate (thus possibly more expensive) numerical simulation methods does not efficiently improve convergence. We also present an upper bound on the error norm and an example to illustrate our results.

Novel pressure-equilibrium and kinetic-energy preserving fluxes for compressible flows based on the harmonic mean

Employing physically-consistent numerical methods is an important step towards attaining robust and accurate numerical simulations. When addressing compressible flows, in addition to preserving kinetic energy at a discrete level, as done in the incompressible case, additional properties are sought after, such as the ability to preserve the equilibrium of pressure that can be found at contact interfaces. This paper investigates the general conditions of the spatial numerical discretizations to achieve the pressure equilibrium preserving property (PEP). Schemes from the literature are analyzed in this respect, and procedures to impart the PEP property to existing discretizations are proposed. Additionally, new PEP numerical schemes are introduced through minor modifications of classical ones. Numerical tests confirmed the theory hereby presented and showed that the modifications, beyond the enforcement of the PEP property, have a generally positive impact on the performances of the original schemes.

Accurate numerical simulations of capillary underfill process for flip-chip packages

Accurate numerical simulations of capillary underfill process for flip-chip packages